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# Induced current in a wire

## Video transcript

let's say I have a magnetic field popping out of this video so these little brown circles show us the the tips of the vectors popping out of this of our screen and then in that magnetic field I have this wire this off-white colored wire and sitting on that off-white colored wire I have a charge of charge Q and let me write down the other stuff so this is a magnetic field B coming out and let's take let's say I were to take this whole wire and let's say that the wire overlaps with the magnetic field a distance of L so let's say the magnetic field stops here and stops here and let's say that this distance right here that distance is L I drew it a little bit weird but you get the idea from here to here is L I have this charge sitting on this some type of conductor that we can consider a wire and the magnetic field is pushing out of the page so in this current formation let's say you know I don't have any voltage across this wire or anything what's going to happen well if I just have a stationary charge sitting in a magnetic field nothing really is going to happen right because we know that the force due to a magnetic field the force due to a magnetic field is equal to the charge times the cross product of the velocity of the charge and the magnetic field if this wire is just stationary there's no voltage across the exoteric sorry the velocity of this charge is going to be zero so the velocity is zero we know that you know the magnitude of a cross product is the same thing as so Q is you know that's just a scalar quantity so that's just Q times the magnitude of the velocity times the magnitude of B times sine theta and in these situations where what you know ever anything that's going on in this plane is going to be perpendicular to this magnetic field so the angle between the magnetic field and any velocity if there were any width in this plane would be 90 degrees so you wouldn't have to worry about the sine theta too much but we see if the velocity is zero or the speed is zero the magnitude of the velocity is zero the there's not going to be any net force due to the magnetic field on this charge and nothing interested in going to happen but let's do a little experiment what happens if I were to move this wire if I were to shift it to the left with the velocity V so I take this wire and I shift it to the left whoops yet with the velocity V alright so the whole wire shifting to the left well the whole wire shifting to the left this charge is sitting on that wire right so that charge is also going to move to the left with the velocity V and now things get interesting the charge is moving to the left with the velocity V so now we can apply the first magnetism formula that we learn we can we can apply this formula so what's going to happen to this charge well the force of the charge is going to be the charge times the magnitude of the velocity cross this the magnetic field vector but what's what's it so we know that it is going to be some net force this is nonzero now and this is nonzero we are assuming and we're assuming the charge is nonzero so what direction is the force going to be in so let's do our right hand rule on the cross product V cross B will give us the direction so point your index finger in the direction of the velocity and I have to look at my own hand to make sure I'm doing it right so you point your index finger in the direction of the velocity point your middle finger in the direction of the magnetic field the magnetic field is popping out of the page so your middle finger is actually going to be popping out of the page your next two fingers are just going to do something like that so you're kind of approximating like you're shooting a gun and then what's your thumb going to do your thumb is going to point straight up so this is your this is the palm of that's your thumb this is this could be your nail fingernail figuring out your thumb finger nail of your middle finger right this is the direction of the velocity suitable color the velocity is that way the magnetic field is popping out of the page so the force create on the particle on this charged particle or on this on this charge due to the magnetic field is going to go in the direction of your thumb so the direction of the force is in this direction so the force so what's going to happen there's going to be a net force in this direction on the charge and the charge is going to move upwards right so what what I mean when you start having a moving you can imagine also then you had a multiple charges right if you had multiple charges here and you're moving the whole wire all of those charges are going to be moving upwards and what is another way to call a bunch of moving charges along a conductor well it's a current depending on how much charge is moving per second so at least in a very in very qualitative terms you see that when you move a wire through magnetic field or when you move a magnetic field past a wire right because they're kind of the same thing it's all about the relative motion but if you move a wire through a magnetic field is actually going to induce a current in the wire and we can actually it's going to induce the current in the wire and actually this is how electric generators are generated and I'll do a whole series of videos on on how you you know if you have a using coal or steam or hydro hydropower how that turns essentially that turns these generators around and it induces current and that's how we get electricity from all of these you know various energy sources that essentially just make turbines turn anyway let's go back to what we were doing so let me ask you a question if this if this particle and this all has a point if this particle starts it like kind of the beginning let's say the particle is right here so it starts right where the magnetic field starts affecting the wire and how much work is going to be done on the particle by the magnetic field well what's work work is equal to work is equal to Force Times distance where the force has to be in the same direction as the distance right Force Times distance I won't mess with the vectors right now but they have to be the same direction so how much work is going to be done on this particle so the work is going to be the net force exerted on the particle times the distance well this distance is L right we say once the particle gets here there's no magnetic field up here so it'll the magnetic field will stop acting on it so the total work done work which is equal to force times distance is equal to so the net force is this up here Q I'll leave some space V cross B times the distance and the distance right here is just a scalar quantity so we could put it out front right Q times L times V cross B right this is QV cross B is the force times the distance that's just the work done now how much work is being done per charge right this this is how much work is being done on this charge but let's say you know there might have been multiple charges we just want to know how much work is done per charge so work per charge we could divide both sides by charge so work for charge is equal to this perch are j-- so it is equal to the distance times the velocity that you're pulling the wire to the left with cross the magnetic field this is where it gets interesting so what is work for charge the units of work are energy right joules and charge that's in coulombs so what is the what it what are joules joules per Coulomb joules per Coulomb this is equal to volts volts are joules per Coulomb so this this particle or these charges are going to start moving in this direction as if there is a voltage difference as if there is a potential difference between this point and this point as if this the you know this is a positive voltage terminal and this is the minus voltage terminal so there's actually going to be a voltage or perceived voltage difference between this point at this point that will start making the current flow let's say you didn't even know that there was a magnetic field here you would just see this current flowing we were like oh well there has to be a voltage difference there right but when we're dealing with this because you know when we talk about voltages that was like a potential difference that's something that that that a particle or charge has a higher potential energy and that's why it's moving but it's hard to at least for these purposes say well you have a higher potential energy here it's hard to it's really being created by the magnetic field so in this context people have said that instead of saying that this is creating a voltage difference between this point and this point that the magnetic field on the moving wire is causing that people say that it's creating an electro-motive force or an EMF but EMF the units are still joules per Coulomb or volts and it really is in every way when you're analyzing the circus still the same thing as a potential difference or as a voltage difference but since it seems a little bit more proactive it seems like this magnetic field is actually impacting a force on this wire that is causing the current to move we call it EMF right so we could say that the EMF the electro-motive force or the voltage across from here to here but that they're really the same thing is equal to the distance of the wire that's in the magnetic field times the velocity that you're pulling the wire in cross the magnetic field so let's say I don't let's just throw out a bunch of numbers let's say that the magnetic field is I'll make it easy to Teslas my velocity to the left is 3 meters per second and let's just for fun let's forgive this a little bit of a resistance so we can figure out something so let's say this resistance is I don't know let's say it is it is 6 ohms there's a 6 ohm resistor here so the resistance of the wire from here to here is 6 ohms all wires have some resistance so first of all what's the EMF oh and let's say that this total distance right here let's say that total distance is 12 12 meters so the EMF for induced on the or the electro-motive force put on to the wire by the magnetic field is going to equal the distance of the wire in the magnetic field 12 meters times well when we're just taking the cross product we know that the velocity is perpendicular to the magnetic field so we don't have to worry about sine theta coz theta is already 90 degrees so we just have to worry about the magnitudes so it's going to be 12 meters times the velocity which is 3 meters per second times the magnetic field or the mass due to the magnetic field that's to Teslas and so the EMF is 12 times 3 times to 12 times 6 which is 72 you could say 20 72 volts or 72 joules per Coulomb and now you have that potential difference or that EMF across a 6 ohm resistor right so that's you just go back to voltage is equal to IR or you could write EMF is equal to IR so EMF divided by resistance so if we take this EMF and we divide it by the resistance divided by 6 ohms we get the current right EMF divided by resistance is equal to current so you divide 72 volts or 72 joules per Coulomb divided by 6 ohms and then you get a current going along the this wire right here due to the EMF due to the magnetic field I know it's very messy at this point of 12 amperes anyway I'm all out of time I'll see you in the next video